Overview
This study introduces an identity-based framework for sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg (CKN) inequalities. These inequalities are associated with the anisotropic radial derivative $\mathcal R_K(u)(x)=\frac{x\cdot\nabla u(x)}{\hK{x}}$, where $K\subset \RN$ is a convex body containing the origin in its interior, and $\hK{\cdot}$ is its Minkowski functional. A central aspect of this work is that $K$ is not assumed to be origin-symmetric, which implies that the Minkowski functional $\hK{\cdot}$ may not be even. This relaxes a common assumption in traditional norm-based anisotropic arguments.
Research Context
The existing $L^2$-Caffarelli-Kohn-Nirenberg theory often relies on the assumption of origin-symmetric convex bodies, allowing for the use of even, norm-based Minkowski functionals. When $K$ is not origin-symmetric, the Minkowski functional $\hK{\cdot}$ is not necessarily even, which prevents direct application of these conventional anisotropic arguments. The present work aims to extend the sharp $L^2$-CKN theory to this more general setting, accommodating convex bodies containing the origin in their interiors without requiring origin-symmetry.
Approach
The research develops an identity-based framework. This framework utilizes anisotropic $L^2$-Hardy and $L^2$-Caffarelli-Kohn-Nirenberg identities. Crucially, these identities incorporate explicit nonnegative remainders, which are key to deriving the sharp inequalities. The methodology focuses on understanding how the best constants of these inequalities depend on the parameter region of $(a,b)\in\mathbb R^2$. The study also investigates the attainability of these sharp constants within a natural completion space and identifies the corresponding extremal functions.
The approach can be characterized by the following key elements:
- Development of an identity-based framework for anisotropic $L^2$-CKN inequalities.
- Introduction of the anisotropic radial derivative $\mathcal R_K(u)(x)=\frac{x\cdot\nabla u(x)}{\hK{x}}$.
- Consideration of convex bodies $K$ where the Minkowski functional $\hK{\cdot}$ is not necessarily even.
- Derivation of anisotropic $L^2$-Hardy and $L^2$-CKN identities that include explicit nonnegative remainders.
- Analysis of the best constants' dependence on the parameter region $(a,b)\in\mathbb R^2$.
- Investigation into the attainability of sharp constants in a completion space.
- Identification of extremal functions.
Findings
The study yielded several findings:
- A new identity-based framework explicitly enables the study of sharp anisotropic $L^2$-CKN inequalities for convex bodies where the Minkowski functional may not be even.
- The derived sharp anisotropic $L^2$-CKN inequalities possess best constants whose values are contingent on the specific parameter region of $(a,b)\in\mathbb R^2$.
- The sharp constants are attainable in a natural completion space, and the corresponding extremal functions were obtained.
- As direct consequences, sharp anisotropic Heisenberg-type uncertainty principles were derived.
- Max-type anisotropic gradient inequalities were also deduced.
- When $K$ is set as the Euclidean unit ball, the results obtained recover the classical Euclidean $L^2$ theory.
- When $K$ is origin-symmetric, the present results align with the usual norm-based anisotropic framework.
- The work extends the sharp $L^2$-Caffarelli-Kohn-Nirenberg theory to general convex bodies containing the origin in their interiors, including cases where the Minkowski functional is non-even.